Path Equivalence

Definition

Two paths $f,g:I\to X$ with the same endpoints $f(0)=g(0)=x_0$ and $f(1)=g(1)=x_1$ are path equivalent (or homotopic relative to endpoints) if there exists a continuous map $H:I\times I\to X$ such that:

1. $H(s,0)=f(s)$
2. $H(s,1)=g(s)$
3. $H(0,t)=x_0$
4. $H(1,t)=x_1$

This relation defines an equivalence relation on the set of paths from $x_0$ to $x_1$. We denote the equivalence class of a path $f$ as $[f]$. In the context of the fundamental group, we consider the specific case where $x_0=x_1$.

References

hatcher2001-algebraic-topology munkres2000-topology